Motion with constant acceleration is an important special case which we now shall treat mathematically.
Here we shall use the following mathematical results about differentiating
polynomia:
We saw that the parabolic x(t) obtained in our inclined track
experiment corresponds to motion with constant
acceleration. It turns out that any motion with
constant acceleration has a parabolic form for x(t).
It is easy to show this. We write the most general form for a
parabola:
I have chosen suggestive names for the three constants which fully
define the motion. For now we confirm that
By differentiating once we derive the velocity versus time:
This establishes that 
is the velocity at time t=0.
By differentiation the second time we obtain the acceleration:
Thus we confirm that Eq. 2 indeed describes
motion with constant acceleration.
Now let's look at some examples of how to apply these
formulae to analyze motion with constant acceleration.
As we do so we should keep in mind that
three constants and no more are required to specify this type
of motion. What is sometimes tricky here is to identify the
three required pieces of information in a given problem
which establish the values for 
, and a.